Conventionally, various services such as Internet banking and electronic applications with administrative agencies have been provided using telecommunication circuits such as the Internet.
To use such services, an authentication process is required to ensure that users of the services are not spoofers or fictitious persons but are correct users. Thus, an electronic authentication technique based on public key cryptography using a public key and a secret key has been frequently employed as a highly reliable authentication method.
Recently, an authentication system using ID-based encryption or a group signature has been developed in order to easily and efficiently manage more users.
In the ID-based encryption or group signature, a necessary exponentiation or scalar multiplication is performed together with a pairing computation. These computations are required to be performed at a high speed in order to shorten the time necessary for the authentication process as much as possible.
Therefore, developed is a technique of enhancing the speed of such exponentiation or scalar multiplication by using a binary method, a window method, or other methods.
Moreover, developed is a technique of enhancing the speed of scalar multiplication by reducing the number of computations using mapping (see Patent Document 1 and Patent Document 2, for example).
Patent Document 1: Japanese Patent Application Publication No. 2004-271792
Patent Document 2: Japanese Patent Application Publication No. 2007-41461